Which of the following statements (s) is/are TRUE ?
I. `11 (1)/(2) + 17 (3)/(4) - 5 (1)/(5) - 2 (1)/(10) = (439)/(20)`
II. `(9)/(1078) gt (11)/(1127) gt (12)/(1219)`
III. `(149)/(151) gt (153)/(155) gt (157)/(159)`

Options

A. Only I
B. Only II
C. Only III
D. None is true

Answer

To determine which of the statements is true, we will analyze each statement step by step.

### Statement I:
**Check: \( 11 \frac{1}{2} + 17 \frac{3}{4} - 5 \frac{1}{5} - 2 \frac{1}{10} = \frac{439}{20} \)**

1. Convert mixed fractions to improper fractions:
- \( 11 \frac{1}{2} = 11 + \frac{1}{2} = \frac{22}{2} + \frac{1}{2} = \frac{23}{2} \)
- \( 17 \frac{3}{4} = 17 + \frac{3}{4} = \frac{68}{4} + \frac{3}{4} = \frac{71}{4} \)
- \( 5 \frac{1}{5} = 5 + \frac{1}{5} = \frac{25}{5} + \frac{1}{5} = \frac{26}{5} \)
- \( 2 \frac{1}{10} = 2 + \frac{1}{10} = \frac{20}{10} + \frac{1}{10} = \frac{21}{10} \)

2. Combine the fractions:
\[ \frac{23}{2} + \frac{71}{4} - \frac{26}{5} - \frac{21}{10} \]

3. Find the least common multiple (LCM) of the denominators (2, 4, 5, 10), which is 20.

4. Convert each fraction to have a denominator of 20:
- \( \frac{23}{2} = \frac{23 \times 10}{2 \times 10} = \frac{230}{20} \)
- \( \frac{71}{4} = \frac{71 \times 5}{4 \times 5} = \frac{355}{20} \)
- \( \frac{26}{5} = \frac{26 \times 4}{5 \times 4} = \frac{104}{20} \)
- \( \frac{21}{10} = \frac{21 \times 2}{10 \times 2} = \frac{42}{20} \)

5. Substitute back into the equation:
\[ \frac{230}{20} + \frac{355}{20} - \frac{104}{20} - \frac{42}{20} \]

6. Combine the fractions:
\[ \frac{230 + 355 - 104 - 42}{20} = \frac{439}{20} \]

**Conclusion for Statement I:** True

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### Statement II:
**Check: \( \frac{9}{1078} > \frac{11}{1127} > \frac{12}{1219} \)**

1. To compare the fractions, cross-multiply:
- For \( \frac{9}{1078} \) and \( \frac{11}{1127} \):
\[ 9 \times 1127 \quad \text{and} \quad 11 \times 1078 \]
Calculate:
- \( 9 \times 1127 = 10143 \)
- \( 11 \times 1078 = 11858 \)

Since \( 10143 < 11858 \), \( \frac{9}{1078} < \frac{11}{1127} \).

2. Now compare \( \frac{11}{1127} \) and \( \frac{12}{1219} \):
- Cross-multiply:
\[ 11 \times 1219 \quad \text{and} \quad 12 \times 1127 \]
Calculate:
- \( 11 \times 1219 = 13409 \)
- \( 12 \times 1127 = 13524 \)

Since \( 13409 < 13524 \), \( \frac{11}{1127} < \frac{12}{1219} \).

**Conclusion for Statement II:** False

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### Statement III:
**Check: \( \frac{149}{151} > \frac{153}{155} > \frac{157}{159} \)**

1. Compare \( \frac{149}{151} \) and \( \frac{153}{155} \):
- Cross-multiply:
\[ 149 \times 155 \quad \text{and} \quad 151 \times 153 \]
Calculate:
- \( 149 \times 155 = 23095 \)
- \( 151 \times 153 = 23103 \)

Since \( 23095 < 23103 \), \( \frac{149}{151} < \frac{153}{155} \).

2. Now compare \( \frac{153}{155} \) and \( \frac{157}{159} \):
- Cross-multiply:
\[ 153 \times 159 \quad \text{and} \quad 155 \times 157 \]
Calculate:
- \( 153 \times 159 = 24387 \)
- \( 155 \times 157 = 24385 \)

Since \( 24387 > 24385 \), \( \frac{153}{155} > \frac{157}{159} \).

**Conclusion for Statement III:** False

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### Final Conclusion:
- Statement I is **True**.
- Statement II is **False**.
- Statement III is **False**.